Question

prove that the line lx +my +n =0 will touch the parabola y^2=4ax if ln =am^2​

Answer 1

Answer:

The equation of the given line is

lx + my + n= 0

and,

x = -1/l(my+n)

Now,

Putting this value of x in the equation of parabola to find the intersection points.

We get,

y^2 = -4a/n(my+n)

or,

ly^2 + 4amy + 4an = 0

Now, for the given line to only touch the parabola and not intersect the root of this quadratic equation In should be repeated twice so that we don't get two values in y.

This can be done when the discriminant of this equation is equal to zero.

Therefore,

(4am)^2 - 4(l)(4an) = 0

or,

16 a^2 m^2 = 16 aln

or,

am^2 = ln

Hence proved.